Random clouds and an application to censoring in survival analysis

Estimation of a distribution under generalized censoring

“…The proof follows virtually the same sort of arguments as in [3], although the underlying structure is slightly different.…”

“…(This last equality requires that ξ be a domain: see [3], page 267). Therefore, {C 1/m ∩ ξ = ∅} ∈ F , and the lemma follows.…”

“…In recent papers, [1][2][3], the problem of survival analysis on a general partially ordered complete separable metric space was approached from the point of view of set-indexed martingales (cf. [4]).…”

Hazard Estimation under Generalized Censoring

“…The framework we will be using is very similar to that used in [6], [8] and [9], except for some details. Let T be a compact complete separable metric space and λ a finite measure on B, the Borel sets of T .…”

“…A simple example is T = [0, 1] d , and we will return to this space in the sequel. Preserving the notation in [9], if D is an arbitrary class of sets, then D(u) will denote the class of finite unions of sets from D. For D an arbitrary subset of T , let T D be a countable dense subset of D. We will use '⊂' to indicate strict inclusion; moreover, D denotes the closure and D • denotes the interior of D. All processes will be indexed by sets belonging to an indexing collection A: Definition 2.1 A nonempty class A of compact, connected subsets of T is called an indexing collection if it satisfies the following:…”